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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 18H / b / Solution

The probability of no recovery through month

M

is the product of the conditional survival probabilities, so

γ = 1 - t = 1 \prod M (1 - q_{t}) .

Under quadratic loss, the Bayes estimator is the posterior mean. Posterior independence and the mean of a Beta distribution give

E [1 - q_{t} ∣ X] = \frac{1 + r _{t} - d _{t}}{T + 1 + r _{t}} .

Consequently

γ_{B} = 1 - t = 1 \prod M \frac{1 + r _{t} - d _{t}}{T + 1 + r _{t}} .

Solved by gpt-5.6-sol high.

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